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Journal Pre-proof A SHORELINE EVOLUTION MODEL CONSIDERING THE TEMPORAL VARIABILITY OF THE BEACH PROFILE SEDIMENT VOLUME (SEDIMENT GAIN / LOSS)
Camilo Jaramillo, Martínez-Sánchez Jara, Mauricio González, Raúl Medina PII:
S0378-3839(19)30203-0
DOI:
https://doi.org/10.1016/j.coastaleng.2019.103612
Reference:
CENG 103612
To appear in:
Coastal Engineering
Received Date:
03 May 2019
Accepted Date:
22 November 2019
Please cite this article as: Camilo Jaramillo, Martínez-Sánchez Jara, Mauricio González, Raúl Medina, A SHORELINE EVOLUTION MODEL CONSIDERING THE TEMPORAL VARIABILITY OF THE BEACH PROFILE SEDIMENT VOLUME (SEDIMENT GAIN / LOSS), Coastal Engineering (2019), https://doi.org/10.1016/j.coastaleng.2019.103612
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Journal Pre-proof A SHORELINE EVOLUTION MODEL CONSIDERING THE TEMPORAL VARIABILITY OF THE BEACH PROFILE SEDIMENT VOLUME (SEDIMENT GAIN / LOSS) Camilo Jaramillo1*, Martínez-Sánchez Jara1, Mauricio González1, Raúl Medina1 1Environmental
Hydraulics Institute, Universidad de Cantabria - Avda. Isabel Torres, 15, Parque Científico y Tecnológico de Cantabria, 39011, Santander, Spain
* Corresponding author Email addresses: [email protected] (Camilo Jaramillo), [email protected] (MartínezSánchez Jara), [email protected] (Mauricio González), [email protected] (Raúl Medina)
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Journal Pre-proof ABSTRACT The prediction of shoreline variability along sandy coastlines is a valuable skill for a broad range of coastal engineering applications. Currently, there are multiple proposals in the literature for equilibrium cross-shore shoreline evolution models based on a constant equilibrium condition correlated with the shoreline position. The aim of this paper is to present an extension of an existing equilibrium shoreline evolution model adding a rate of change component, which in turn modifies the relationship between the equilibrium shoreline position and the incident wave energy, as a function that advances or retreats over time. The proposed model is applied to two study sites on the Spanish coast. On the one hand, one and a half years of shoreline position data obtained from a video camera system at Nova Icaria Beach, Catalonia, are used to evaluate the shoreline evolution during a period of sediment contribution, which is attributed to a sand bypass coming from Bogatell Beach. On the other hand, 16 years of shoreline position data acquired from beach profile surveys in Campo Poseidón, Huelva, are used to reproduce the shoreline evolution, taking into account the trend of sediment loss due to sediment drift alongshore. The results obtained from both study sites using the proposed model show overall good performance. The model successfully represents the general shoreline erosion–accretion oscillations and the progressive trend of sediment gains or losses through the evolutionary relationship between the equilibrium shoreline position and the incident wave energy.
Key words: equilibrium shoreline evolution models, cross-shore, shoreline trend, equilibrium energy function.
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Journal Pre-proof 1. INTRODUCTION Most sandy beaches worldwide have a tendency of progressive shoreline advances or retreats for years or even decades, and this pattern is a product of multiple factors, which may be of natural or anthropic origin. In addition, the beaches usually suffer from specific alterations at a shorter time scale, which are reflected by a representative sediment deficit or surplus (Mentaschi et al., 2018). Therefore, to predict the shoreline variability in such cases, the numerical models of morphological evolution must contemplate a rate of change of shoreline position for sediment gain or loss. In the literature, there is an extensive list of different types of models that simulate shoreline evolution: multi-line shoreline models (e.g., Bakker, 1970; Perlin and Dean, 1979; 1983; Hanson and Larson, 2000), one-line shoreline models (e.g., Pelnard-Considere, 1956; Hanson and Kraus, 1991; Dabees and Kamphuis, 1998), equilibrium shoreline evolution models (e.g., Kriebel and Dean, 1993; Yates et al., 2009; Davidson et al., 2010; 2013; Jara et al., 2015) or combined models (e.g., Vitousek et al., 2017; Robinet et al., 2017; 2018). The computational cost of the processbased morphodynamic models (van Rijn et al., 2003) prevents their application to multi-annual or even inter-annual morphological changes. In contrast, equilibrium shoreline evolution models are computationally efficient and can be applied to investigate long-term morphological changes (Davidson and Turner, 2009). The equilibrium shoreline evolution models try to reproduce the shoreline evolution based on a kinetic equation in which the change rate is proportional to the difference between a theoretical equilibrium condition and the current conditions. Recently, Jara et al., (2018) presented a summary of most up-to-date literature related to this type of model, and the summary was divided into two approaches: 1. models based on an equilibrium condition correlated with the shoreline position (e.g., Yates et al., 2009; Jara et al., 2015; Doria et al., 2016) and 2. models based on an equilibrium condition as a weighted average of antecedent conditions (e.g., Davidson et al., 2013; Splinter et al., 2014). The model proposed in this paper is framed into the first group. Most likely, the most widely used model based on an equilibrium condition correlated with the shoreline position corresponds to the proposal of Yates et al., 2009 (hereafter defined as the YA09 model). The YA09 model has been successfully applied to various beaches governed by strong seasonal and interannual variability due to cross-shore sediment transport (e.g., Yates et al., 2011; Castelle et al., 2014; Lemos et al., 2018); however, these study cases did not experience a marked trend of sediment gains or losses in the medium-long term. To extend the application of the empirical equilibrium models and to account for unexplained shoreline motions, such as long-term trends that are uncorrelated with wave climate changes at small time scales (days or weeks), some authors have added a constant to the model equation. In this way, Long and Plant, (2012) proposed a shoreline change model incorporating short- and long-term evolution integrated into a data assimilation framework. They considered a linear trend term for longshore approximation; nevertheless, they followed the YA09 bases, assuming a constant relationship between the equilibrium shoreline position and the incident wave energy (this relation is known as the “equilibrium energy function”, EEF; Jara et al., 2015) for forecasting purposes. Long and Plant, (2012) tested their model in a synthetic case of 2 years of monthly sampled data, but not in any cases of real beaches. Jara et al. (2015) concluded that the relationship between the equilibrium shoreline position and the incident wave energy is implicitly related to the morphological beach characteristics; 3
Journal Pre-proof more specifically, a constant linear relationship would be associated with a constant sediment volume delimited in the active beach profile. The above relation is translated into the following consequence: if an empirical equilibrium model uses a constant EEF for a study site with a net sediment volume deficit or surplus, then the evolution model would amplify or overestimate the accretion-erosion oscillations in an unrealistic way. Therefore, a constant EEF is not appropriate for study sites subjected to significant sediment gains or losses. Based on the above, the aim of this study is to present a new extension of the YA09 model; this extension considers the variation in the EEF by means of a trend rate of sediment gain or loss, 𝑣𝑙𝑡. This rate is a source or sink of sediments related to unresolved processes, as defined by Vitousek et al., (2017). There are multiple natural or anthropic origins that may cause this trend rate on sandy beaches, such as littoral sediment drift alongshore (e.g., Komar and Inman, 1970; Miller, 1999), sediment discharge by the rivers (e.g., Baban, 1995; Boateng et al., 2012), nourishments (e.g., Hanson et al., 2002; Ludka et al., 2018; Tonnon et al., 2018), dune erosion (e.g., Castelle et al., 2017; Splinter et al., 2018), cliff retreat (e.g., Limber et al., 2018; Young, 2018), aeolian sediment transport (e.g., Davidson-Arnott and Bauer, 2009; Cohn et al., 2018) and others. This paper is organized as follows: first, the theoretical development of the proposed model is explained in Section 2. Two study sites were selected to test the model: first, we used one and a half years of shoreline position data obtained from a video camera system at Nova Icaria Beach, Catalonia; and second, we used approximately 20 years of shoreline position data acquired from beach profile surveys at Campo Poseidón, Huelva. The study sites, marine conditions and shoreline acquisition methods are detailed in Section 3. The model results for both study sites are shown in Section 4, and further discussions are provided in Section 5. Finally, the main conclusions are summarized in Section 6. 2. MODEL DEVELOPMENT As stated previously, several shoreline evolution models based on an equilibrium condition correlated with the shoreline position have been developed to simulate the shoreline position variability due to storm response and seasonal changes. These models relate the rate of cross𝑑𝑆 shore shoreline displacement, 𝑑𝑡, to the incident wave energy and the wave energy disequilibrium between the wave energy and the equilibrium wave energy that would cause no change to the present shoreline location (Castelle et al., 2014). The model proposed in the present study is based on the kinetic equation defined by Yates et al. (2009). According to the YA09 model, the governing equation that describes the rate of shoreline change is of the form (Eq. (1)): 1 𝑑𝑆(𝑡) = 𝐶 ± 𝐸 2 ∙ ∆𝐸(𝑆) 𝑑𝑡
(1) ∆𝐸 = (𝐸 ― 𝐸𝑒𝑞) (2) where S(t) and E are the shoreline position (m) and the incident wave energy (m2), respectively, at time “t”, E1/2 is the square root of the incident wave energy as an energy weighting factor that 4
Journal Pre-proof prevents nonphysical changes in the shoreline position (Yates et al., 2009), ΔE (see Eq. (2)) is the wave energy disequilibrium, C± is the proportionality constant of accretion (m-2s-1), C+, when ΔE < 0, or erosion, C-, when ΔE > 0, and Eeq is the equilibrium wave energy. The shoreline position, S, is computed as the average position of the shoreline in relation to a constant reference point or line in the backshore (see Fig. 2). Following Jara et al., (2018), the incident wave energy, E can be expressed as the zero-order moment of the wave height at breaking, Hb, according to Thornton and Guza, (1983). 𝐸=
𝐻𝑏
( )
2
4.004
(3) In both formulations (Eq. (1) and (2)), it remains to be defined how the equilibrium wave energy, Eeq, is estimated. The value of Eeq is determined in the YA09 model using a linear function of the shoreline position following the next expression: 𝐸𝑒𝑞(𝑆) = 𝑎𝑆 + 𝑏 (4) where “a” and “b” are the slope (m2/m) and the ordinate axis intercept (m2), respectively. Yates et al. (2009) defined the equilibrium shoreline position, 𝑆𝑒𝑞, for a given wave energy as follows: 𝑆𝑒𝑞 =
𝐸―𝑏 𝑎 (5)
The analytical correlation of Eq. (5) is the previously defined EEF (Jara et al., 2015). Basically, this correlation is the cornerstone of shoreline evolution models based on an equilibrium condition correlated with the shoreline position. As Jara et al. (2015) concluded, the EEF implicitly considers the sediment volume available in the physiographic unit; therefore, a constant EEF is useful for beaches that preserve their sediment volume over time. However, a constant EEF is not useful for study sites subjected to significant sediment gains or losses. Based on the aforementioned aspects, we propose an extension of the YA09 model including a linear trend term, 𝑣𝑙𝑡, for the longshore approximation (m/s) and this term should be considered in the EEF behavior while the beach under study is subjected to net sediment gains or losses. In this sense, Eq. (5) must evolve over time in such a way that the relationship is an increasing or decreasing function, depending on the rate of sediment gain/loss acting on the physiographic unit. In this study, the EEF takes the following form:
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𝐸―𝑏 + 𝑣𝑙𝑡 ∙ 𝑡 𝑎 (6)
It is important to note that Long and Plant, (2012) (hereafter referred to as the LP12 model), included the linear trend term, 𝑣𝑙𝑡, to the YA09 model, but in the Eq. (1) adding also the Kalman filter. This data assimilation method automatically adjusts the model parameters (a, b, C± and 𝑣𝑙𝑡 ) during runtime to best fit any available observed shoreline data at the concurrent time step (Vitousek et al. 2017). In other words, as concluded by Vitousek et al. (2017), the Kalman filter is used as a largescale calibration tool, and after the last available shoreline observation is assimilated, the model parameters remain constant for the entire forecasting period. From the above, it is inferred that the LP12 model, including the Kalman filter, uses an EEF that evolves during the calibration period, but then, it remains constant for the forecasting period. Fig. 1 shows an EEF example of the proposed model based on the study site of Nova Icaria Beach, which is evaluated in detail in Section 4.1. The upper panel of Fig. 1 shows only one month of time series of incident wave energy, and the bottom panel shows the corresponding empirical approximation of the EEF (black line). The points are consecutive measurements that are colored according to the shoreline change rate.
Fig. 1. One month of incident wave energy at Nova Icaria Beach (upper panel) as driver of an EEF (black line in bottom panel). The points are consecutive measurements that are colored according to the shoreline change rate.
2.1. Model hypotheses and assumptions The proposed model in this study assumes the same bases as the YA09 model, in terms that shoreline changes mainly respond to wave energy and it is insensitive to wave direction, and as such, it is best suited for locations where waves are the primary driver of shoreline response.
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Journal Pre-proof Also, the model does not account for short-scale processes such as beach cusp formation, alongshore variable bar welding, or rip current embayments (Splinter et al., 2014). By neglecting the wave directionality of each sea state, the model is not able to capture the short-term variability of the longshore sediment transport. However, the long-term shoreline change due to gradients in longshore transport and/or onshore/offshore feeding/loss of sand may be captured by the linear trend term, 𝑣𝑙𝑡. This linear term has been included to consider the overall longshore processes, considering that where these processes cannot be captured by the linear term (or the wave-driven model component), the model does not accurately resolve the shoreline response. Following the conclusions found by Yates et al., (2009); (2011) the sediment grain size affect the magnitude of the equilibrium slope “a” and the rate change coefficients C±, both of which affect the rate of beach change. From the above, variations of free parameter values between sites are implicitly associated with the beach physical characteristics. In this study, it is considered that the calibration parameters (C- and C+) associated to erosion and accumulation processes remain constant during the entire model simulation. With respect to the tidal range, the model does not explicitly include any additional parameter. Nonetheless, as suggested by Castelle et al., (2014) the equilibrium shoreline evolution models can be applied to a range of elevation contours in the intertidal zone to evaluate the effect of wave energy at different altitudes along the intertidal beach profile, with satisfactory efficiency. Castelle et al., (2014) concluded that the best shoreline proxy in their mesotidal study site was related to the mean high water level, where the inner-bar and berm dynamics have little influence on the shoreline cross-shore displacement. This conclusion was supported by Lemos et al., (2018) who analyzed a macrotidal environment and found that the equilibrium models shows a good predictive ability in the upper intertidal zone where the sediment dynamics depend mainly on the waves energy. As suggested by Splinter et al., (2014), the exclusion of water level precludes the impacts of changes in mean water level due to climatological impacts, such as storm surge, El Niño Southern Oscillation (ENSO), and sea level rise. 2.2. Model implementation From the governing equations described above, this section presents a brief methodology to apply the proposed model. The first step is to identify whether the beach under study is subjected to a net gain or loss of sediments. For this, the slope of the trend term, 𝑣𝑙𝑡, relies on a regression of historical data as long as possible. This term can be obtained by historical records of shoreline surveys or, for example, by evaluating the average rate of alongshore sediment transport by means of historical wave data propagated to the coast. The next step is to determine the remaining calibration parameters (a, b and C±). For this, it is necessary to define a calibration time period in which a constant EEF is evaluated by means of Eqs. (1), (2) and (6), considering that 𝑣𝑙𝑡 = 0. This initial relationship is the best fit line to the observed average wave energy causing no change in the shoreline position. The calibration parameters C± must be obtained through an iterative algorithm (see Section 4). The calibration period can be considered as follows:
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A period of time in which there is no significant trend of a gain or loss of sediments. A period of time with a significant trend of a gain or loss of sediments. In this case, the trend term, 𝑣𝑙𝑡, must be filtered.
Once all the calibration parameters (a, b, C± and 𝑣𝑙𝑡) have been defined, the model is validated using Eqs. (1), (2) and (6), considering that 𝑣𝑙𝑡 ≠ 0. In summary, the EEF starts from an initial linear relationship from which it evolves according to the rate of sediment gain/loss acting on the beach. 3. STUDY SITES In this section, the selected study sites are presented. The right panel of Fig. 2 corresponds to Nova Icaria Beach, as this represents a study site where the beach accretion process was experienced in a limited time period (Ojeda and Guillén, 2006); on the left panel of Fig. 2, there is a monitored beach profile of Rompeculos Beach (referred to as the Campo Poseidón beach profile), where a continuous sediment loss process has been recorded (Fernández et al., 1990).
Fig. 2. Location of the selected study sites. Campo Poseidón (lower and left panel), including the “Downscaled Ocean Waves” (DOW) position, Tide Gauge Huelva and monitoring beach profile. Nova Icaria Beach (upper and right panel), including the DOW position, Tide Gauge Barcelona and the ARGUS station with the corresponding Camera C5. In both cases the reference point or line to estimate the shoreline position, S, is indicated.
The marine conditions acting in the vicinity of both study sites (see Sections 3.1.2 and 3.2.2) are characterized by the incoming waves acquired from the “Downscaled Ocean Waves” (DOW) (Camus et al., 2013) and the sea levels recorded by the closest tide gauges. The DOW database is a historical reconstruction of coastal waves by means of a hybrid methodology based on both dynamical and statistical downscaling for a period of 68 years (1948 to 2015) along the Spanish coast; this database provides hourly sea states with different wave parameters (e.g., significant wave height, Hs, peak period, Tp, mean wave direction, θm). Specifically, the DOW is a downscaled wave reanalysis of coastal zones from a Global Ocean 8
Journal Pre-proof Waves (GOW) database (Reguero et al., 2012), which considers a correction of open sea significant wave height through a directional calibration (Mínguez et al., 2011) and has been validated with records from the closest buoys. Regarding sea levels, the Spanish Grid of Gauges (REDMAR) provides data for astronomical tide levels and surge tide levels. The selected tide gauges correspond to the Port of Barcelona and the Port of Huelva for Nova Icaria Beach and Campo Poseidón, respectively.
3.1. Nova Icaria Beach 3.1.1. Study site Nova Icaria Beach is a 400 m-long non-barred sandy beach located at Barcelona (Catalonia), northeastern Spain (see Fig. 2). The beach is embedded between the Olympic Port of Barcelona to the south and the Bogatell dike to the north. The beach is also protected by a semi-submerged curved breakwater, which is an extension of the Bogatell dike (Z = 0 m with respect to the MSL), and a small breakwater that protects the boat-access ramp in the northeastern section of the Olympic Port. The average direction normal to the beach orientation is close to 135° with respect to north, and the beach sediment is characterized by an average median grain size (D50) of approximately 0.43 mm (Ojeda and Guillén, 2008). Nova Icaria Beach is one of the most intensively studied coastal sites in Catalonia (e.g., Ojeda and Guillén, 2006; Guillén et al., 2008; Ribas et al., 2010; Ojeda et al., 2011; Turki et al., 2012; 2013; Jara et al., 2015), and its historical morphodynamic variability has been reported in detail. In this study, it is worth highlighting the shoreline monitoring analysis of Barcelona's beaches developed by Ojeda and Guillén (2006) and Guillén et al. (2008), who identified a process of sediment bypass from Bogatell Beach to Nova Icaria Beach. This event occurred months after nourishment works at Bogatell Beach were carried out between 13th June and 5th July 2002, with a sediment contribution of approximately 70,000 m3. It is important to highlight that headland bypassing due to moving sand around Bogatell dike (see Fig. 2) was initially reported by Peña and Covarsl (1994), who used drift monitoring analysis using transects. This sediment drift was controlled in 2010, when the construction of a submerged barrier was implemented to delimit the sand contained in the physiographic unit of Bogatell Beach. 3.1.2. Marine conditions The Catalan coast (Western Mediterranean) is a microtidal region where the astronomical tide is characterized by a semi-diurnal regime (mean range of 0.23 m), and storm surges can reach values up to 0.5 m greater than the astronomically predicted tide level. In this region, the effect of tides on the morphology of the beaches is significantly less than waves. The selected DOW point near Nova Icaria Beach is located at latitude 41.387° and longitude 2.207°, with a water depth of 13.3 m (see Fig. 2). This point has been validated with measurements of an OPPE (Organismo de Puertos del Estado) buoy at Barcelona. Fig. 3a shows the scatter diagram that correlates the significant wave height, Hs, with the peak period, Tp, and Fig. 3b depicts the directional rose of the Hs.
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Fig. 3. Wave characteristics of DOW point at Nova Icaria Beach. a) Scatter diagram Hs-Tp. b) Directional rose of Hs.
A statistical analysis of the wave conditions from 1948 to 2015 shows that the dominant or more frequent waves and the most energetic waves come from the ESE (24% of the time), followed by waves coming from the E (21.9%) and SE (21.7%). The Hs and Tp have ranges of 0.01–5 m and 1–13.4 s, respectively. The wave height exceeded 50% of the time is 0.41 m, and the significant wave height exceeded 12 hours per year (Hs12) is approximately 3 m. The significant wave height time series presents a cyclical behavior, with storm periods (October - April) separated by periods of low storm activity (May - October). 3.1.3. Shoreline data Daily shoreline records during the time period, which spanned from May 2002 to October 2003, were obtained from snap images taken by the Argus video system “Barcelona Littoral Station” located on top of the Mapfre Building by the Olympic Marina approximately 142 m above mean sea level (MSL). This Argus system consists of five video cameras operated by the Mediterranean Center for Marine and Environmental Research (CMIMA), and these cameras have a 180° view of the coast. The shorelines were manually plotted and digitized over oblique pictures taken by camera C5 and then rectified by means of a direct linear transformation technique, with a maximal expected error of 1 m at the northern tip of the beach (Jara et al. 2015). Fig. 4a shows an example of a rectified snap picture including two shorelines associated with the summers before and after the bypass period (approximately from October 2002 to June 2003) from Bogatell Beach to Nova Icaria Beach. Fig. 4b shows the time-series evolution of the mean cross-shore position.
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Fig. 4. Shoreline position at Nova Icaria Beach. a) Example of rectified snap picture showing two surveys: the blue line is the shoreline associated with the day 1507//2002 (before bypass), and the red line is the shoreline associated with 15/07/2003 (after bypass). b) Time-series of shoreline positions.
The shoreline position, S, for the entire beach has been computed as the mean position in relation to the constant reference line in the backshore (see Fig. 4a). The shoreline reference level corresponds to the local MSL. To minimize the possible overestimation of shoreline changes due to storm surges and wave set-up, the shoreline positions extracted from rectified pictures have been corrected with the MSL position at the moment the picture was captured, considering the astronomical tide, storm surge and wave set-up. 3.2. Campo Poseidón 3.2.1. Study site The second study site corresponded to the Campo Poseidón beach profile (see Fig. 2). This monitoring section is located at Rompeculos Beach in Moguer municipality (Huelva), on the southwest coast of Spain. It is a coastline stretching 3000 m-long and encompasses a long sandy beach (more than 50 km) bounded by the Port of Mazagón and the delta of the Guadalquivir River. The average direction normal to the Rompeculos Beach orientation is close to 210° with respect to the north, and the beach sediment is characterized by D50=0.3 mm (Fernández et al., 1992). The east coast of Huelva has a long history of monitoring programs, which have shown how highly recessive this area is. Fernández et al. (1990) quantified the coastline retreat at a rate of 1.5 m/yr over 30 years (since the 1960s). This value is consistent with the estimated trend from data measured over more than 16 years (December 1998 to July 2015) at the Campo Poseidón beach profile (see Section 4.2). As Medina et al. (1992) concluded, the coastal retreat of this region has been caused by two different reasons: first, the littoral drift from west to east, and
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Journal Pre-proof second, the reduction in the volume of sand transported by the rivers to the coast mainly caused by human actions (e.g., construction of dams, littoral barriers). 3.2.2. Marine conditions The Gulf of Cadiz faces the Atlantic Ocean and is characterized by a mesotidal regime with a medium neap to spring variation (1.20–3.30 m) (Muñoz-Perez et al., 2001). According to datum references of the Tide Gauge of Port of Huelva, the mean lower low water (MLLW) level is 0.47 m, the MSL is 2.03 m and the mean higher high water level (MHHL) is 3.63 m, all of which are above the reference level or zero of the port. The storm surges can reach values up to 0.6 m greater than the astronomically predicted tide level. The selected DOW point near Campo Poseidón is located at latitude 37.1° and longitude 6.765°, with a water depth of 6.3 m (see Fig. 2). This point has been validated with measurements of OPPE-Sevilla. Fig. 5a shows the scatter diagram Hs-Tp, and Fig. 5b shows the directional rose of the Hs.
Fig. 5. Wave characteristics of DOW point at Campo Poseidón. a) Scatter diagram of Hs-Tp. b) Directional rose of Hs.
The statistical analysis of wave conditions from 1948 to 2015 shows that the most frequent and most energetic waves come from the SW (24% of the time), followed by waves from the S (19%) and WSW (18%). The Hs and Tp have ranges of 0.01–4 m and 1–19 s, respectively. The wave climate is strongly seasonally modulated, with an annual mean significant wave height, Hs, of 0.39 m and an Hs12 of approximately 2.7 m. The winter is the most energetic period (i.e., December-January). 3.2.3. Shoreline data Repsol Investigaciones Petrolíferas S.A. (R.I.P.S.A.) has monitored the Campo Poseidón beach profile since December 1998. From the beginning until 2013, 10 campaigns per year of intertidal beach profile surveys were carried out, and these sampling were monthly except in the months of July and August. From 2013 onwards, the program was extended to 12 measurements per year, including profiles for July and August. Following the research developed by Castelle et al., (2014) and Lemos et al., (2018), the intersection of the coastal profile with the vertical elevation MHHL was selected as a shoreline proxy. According to these authors, the shoreline cross-shore displacement related to the MHHL contour is slightly influenced by the inner-bar and berm dynamics. Based on the above information, Fig. 6 shows the time-series evolution of the mean cross-shore position associated with the MHHL contour.
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Fig. 6. Time-series of shoreline positions at the Campo Poseidón beach profile considering the MHHL contour as the reference level.
4. RESULTS This section presents the performance of the proposed equilibrium cross-shore shoreline evolution model applied to the two selected study sites. In both cases, the time series of waves at the DOW points were propagated to the breaking point using linear wave theory and the Thornton and Guza (1983) linear breaking criterion. The wave breaking conditions have been used to obtain the incident wave energy, which is the main model driver. The calibration procedure consists of finding the best model fit using an error minimizing technique to solve the calibration parameters. To do this, an iteration algorithm was used to find the proportionality constants of accretion, C+, and erosion, C-, that produced the lowest root mean square error (RMSE; Eq. (7)) between the modeled and measured shoreline positions. 𝑛
𝑅𝑀𝑆𝐸 =
∑𝑖 = 1(𝑆𝑑𝑎𝑡𝑎,𝑖 ― 𝑆𝑚𝑜𝑑𝑒𝑙, 𝑖)2 𝑛
(7) where i and N represent each datum and the total size of the sample, respectively. Section 4.1 presents the results from Nova Icaria Beach, and Section 4.2 presents the results from Campo Poseidón. 4.1. Model results from Nova Icaria Jara et al. (2015) analyzed the shoreline evolution of Nova Icaria Beach during a natural beach oscillation time period that spanned from January 2005 to January 2007 using two equilibrium cross-shore shoreline evolution models. They successfully calibrated and validated their proposed model (named the JA15 model) and the YA09 model. From that research, the present study inherits the YA09 model calibration parameters obtained from the six-month period between January and June 2005 (a=-0.068106 m2/m, b=6.13 m2, C+=-2.4·10-4 m-2s-1, C-=-1.1·104 m-2s-1). It should be noted that the optimal erosion and accretion change rate coefficients, C±, are of the same order of magnitude. The proposed model has been validated for a time period of one and a half years (see Fig. 7d). The first and last five months correspond to the natural oscillations of Nova Icaria Beach before and after the sediment bypass period (approximately from October 2002 to June 2003), respectively (Ojeda and Guillén, 2006).
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Journal Pre-proof During the sediment bypass period (gray shaded area in Fig. 7d), the shoreline position of Nova Icaria Beach advanced at a rate of 11.55 m/yr (see Fig. 7a). Fig. 7b shows the EEF that defines the equilibrium shoreline position based on the incident wave energy. It is important to highlight that the beginning of the EEF (to the left in this case) is applicable to Nova Icaria Beach’s situation before the sediment bypass, while the end of the function governs the beach variability after the sediment bypass.
Fig. 7. Results for Nova Icaria Beach. a) Linear fit to the shoreline evolution during the bypass period. b) EEF (black line) considering the complete study period. c) Time series of incident wave energy at the breaking point. d) Validation of the proposed shoreline evolution model.
According to the results presented in Fig. 7d, the proposed cross-shore shoreline equilibrium model including the trend term allows the simulation of the progressive shoreline advance in accordance with the sediment supply contributed during the bypass period from Bogatell Beach to Nova Icaria Beach. In addition, the model exhibits robust behavior when simulating the general shoreline advances and retreats. Both observed and modeled shoreline positions show fast erosion after major storms and slower accretion in the post-storm recovery period. For example, given the time series of incident wave energy at the breaking point (see Fig. 7c), a storm of greater intensity occurred in mid-October 2003; this storm is reflected by a strong shoreline retreat (Fig. 7d). The resulting RMSE between the observed and modeled shoreline positions is 1.62 m, which is similar to the expected error due to the image rectification (1 m at the northern tip of the pocket beach) and to the results obtained by Jara et al. (2015) for the study period of 2005-2007. The correlation coefficient, ρ, between the observed and modeled shoreline positions is 0.87, which is a clear result indicating a strong correlation. 4.2. Model results from Campo Poseidón 14
Journal Pre-proof In the case of the Campo Poseidón beach profile, the first 3 years (December 1998-December 2001) of shoreline measurements were considered for calibration purposes (see gray shaded area in Fig. 8d). Because the beach experiences a continuous retreat at a rate of 1.5 m/yr according to Fernández et al. (1990), the calibration period must filter this trend to obtain the model calibration parameters (a, b and C±). The iteration technique mentioned at the beginning of Section 4 was used to find the best set of free parameters that minimize the RMSE between the modeled and observed shorelines. The resulting calibration parameters are a=-0.0027 m2/m, b=0.5595 m2, C+=-5.7·10-4 m-2s-1 and C-=-9.7·10-5 m-2s-1. Then, the proposed model was validated for more than thirteen years of shoreline measurements (from January 2002 until July 2015), as shown in Fig. 8d. Although the shoreline position record obtained from the beach profile surveys is approximately 20 years (see Section 3.2.3), the simulation was run until mid-2015 based on the wave data availability (see Section 3.2.2). During the study period, the shoreline position at the Campo Poseidón beach profile retreated at a rate of -1.55 m/yr (see Fig. 8a). In this case, the beginning of the EEF (to the right in Fig. 8b) is related to the linear relationship (E-S) defined for the calibration procedure, and then, the EEF evolves based on the sediment loss rate.
Fig. 8. Results at Campo Poseidón. a) Linear fit to the shoreline evolution. b) EEF (black line) considering the complete study period. c) Time series of incident wave energy at the breaking point. d) Calibration (gray shaded period) and validation of the proposed shoreline evolution model.
Fig. 8c shows a strong seasonal variation of the incident wave energy, with strong storms during winter periods and low energy waves during the summers. The results presented in Fig. 8b reveal that the model, again, has achieved good performance, successfully reproducing the general advance/retreat shoreline trend. The resulting RMSE is 10.4 m, and the correlation coefficient,
15
Journal Pre-proof ρ, between the observed and modeled shoreline positions is 0.64, which is a satisfactory correlation. 5. DISCUSSION According to the results presented in Section 4, the proposed model showed significant skill in reproducing the shoreline evolution at two study sites with different characteristics: 1. Nova Icaria Beach was subjected to sediment supply in a microtidal environment, and 2. Campo Poseidón was subjected to sediment loss in a mesotidal environment. In addition, the data sources used to obtain the time series of the shoreline positions were different in both cases; in the first, data came from video-camera images, and in the second, data came from beach profile surveys. The study site of Nova Icaria Beach has been used for direct model validation because the sediment bypass period from Bogatell Beach was defined a priori, and the calibration parameters were inherited from Jara et al. (2015). During the first five months, the model uses a linear EEF with 𝑣𝑙𝑡=0 m/yr; then, the EEF evolves during the bypass period at the rate of sediment increase, and it finally continues with the last linear EFF and 𝑣𝑙𝑡=0 m/yr for the next five months. On the other hand, the study site of Campo Poseidón has been completely predictive, and the model has achieved an overall good performance, simulating nearly 16 years of shoreline evolution. In this case, the value of the loss rate recorded with historical data (𝑣𝑙𝑡 =1.5 m/yr) by Fernández et al. (1990) has been considered, and it has been applied to the longterm in a continuous erosion trend, which results in a recessive EEF. To assess the advantages of the proposed model, Fig. 9 shows a comparative model simulation for the two study sites considering two scenarios: 1. Model includes trend term (same results of Section 4), and 2. Model establishes 𝑣𝑙𝑡=0 m/yr during the entire study period. Table 1 summarizes the values of the mean squared error and correlation coefficient found in this comparative analysis.
Fig. 9. Comparison of results using the proposed model including the trend term versus the model without the trend term at a) Nova Icaria Beach and b) Campo Poseidón.
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Journal Pre-proof Table 1. Root mean square error and correlation coefficient between the observed and modeled shoreline positions at Nova Icaria Beach and Campo Poseidón considering two scenarios.
Nova Icaria Beach 𝒗𝒍𝒕 (m/yr) RMSE (m) ρ Without trend term 5.18 0.08 n/a With trend term 1.64 0.87 11.55 Model
Campo Poseidón 𝒗𝒍𝒕 (m/yr) RMSE (m) ρ 18.7 0.31 n/a 10.4 0.64 -1.5
According to Fig. 9, as expected, when the equilibrium shoreline evolution model does not consider the trend term, it tries to reproduce the shoreline advancing and retreating oscillations around an average position over time; however, the model does not consider net volume changes on the beach. The values presented in Table 1 reflect the weak correlation between the observed and modeled shoreline positions when the model does not include the trend term. Another test that should be highlighted is what would happened if the linear trend term were added after the YA09 model results (similar to LP12 model), but considering that the EEF is constant over time (not including the rate parameter). This assumption is represented in Fig. 10, for the case of Nova Icaria Beach. As it is observed, it would be feasible to find a good result by adding the linear change rate during the bypass period; however, if the model assumes a constant EEF, the simulation will tend to the established equilibrium condition, from which, the model is not able to assimilate that the beach has changed its sediment volume over time.
Fig. 10. Comparison between the proposed model and a simulation considering the linear trend term added after the YA09 model result (bypass period), but assuming that the EEF is constant over time (bottom panel). Differences in meters between both simulations (upper panel).
During the bypass period, only differences less than 25 cm are registered (see upper panel of Fig. 10). These slight variations of erosion/accretion fluctuations are explained in the differences between the kinetic equations of both cases, presented in Table 2. Table 2. Comparison of kinetic formulations between the proposed model versus YA09 model results adding linearly the long term.
YA09 model results adding linearly the long term
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Proposed model
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[
1 𝑑𝑆(𝑡) = 𝐶 ± 𝐸 2(𝐸 ― 𝐸𝑒𝑞(𝑆)) + 𝑣𝑙𝑡 ∙ 𝑡 𝑑𝑡
]
1 𝑑𝑆(𝑡) = 𝐶 ± 𝐸 2(𝐸 ― 𝐸𝑒𝑞(𝑆)) 𝑑𝑡
(10)
(8) 𝐸𝑒𝑞(𝑆) = 𝑎(𝑆 ― 𝑣𝑙𝑡 ∙ 𝑡) + 𝑏
𝐸𝑒𝑞(𝑆) = 𝑎𝑆 + 𝑏
(9)
(11)
It should be noted that the proposed model assumes that 𝑣𝑙𝑡 is linearly constant during the forecasting period; nonetheless, as indicated in the Introduction, in reality, there are multiple sources or sinks of sediments on beaches (e.g., littoral sediment drift alongshore, nourishments, cliff retreat) and they are variable over time. Therefore, assuming a constant rate, the proposed model captures only the general tendency, but the nonlinear changes cannot be precisely resolved. For example, the month of February 2003 in the simulated results of Nova Icaria Beach and the years 2010 and 2012 in the simulated results of Campo Poseidón had more advanced shoreline position records than did the model predictions. Note that in those periods where the model is not able to replicate the shoreline response, it should be also due to the others assumptions described in detail in Section 2.1 (not consideration of wave directionality, changes in the wave-driven model component speed or changes of the physical beach parameters). In order to make the term 𝑣𝑙𝑡 and the other parameters (a, b, C+ and C-) variable over the time, it would be possible to include a data assimilation algorithm, such as the Kalman Filter, previously proposed by Long and Plant, (2012). This algorithm would automatically adjusts the rate parameters during the runtime to best fit any available observed data at the concurrent time step. However, after the last available observation is assimilated, the parameters will remain constant for the entire forecasting period. The limitations inherited by assuming a linear trend constant suggest that the proposed model could even improve its prediction by including a better prediction of the source or sink variability, e.g., a littoral sediment drift alongshore function obtained from historical seasonal analysis or a sediment discharge function associated with streamflow hydrographs of rivers, among others. Finally, time series of shoreline measurements in different sites with diverse beach characteristics and wave conditions would be useful to widely validate the robustness of the shoreline evolution model, including the analytical form of the EEF proposed in this paper. 6. CONCLUSIONS This present study proposes an extension of the equilibrium shoreline evolution model originally developed by Yates et al., (2009). The new model adds a trend parameter due to unresolved processes (constant rate of sediment gain or loss), which in turn modifies the EEF as a pathway that advances or retreats over time. The model is capable of reproducing the shoreline response due to cross-shore forcing over a variety of temporal scales. It is a reduced-complexity empirical evolution model that requires few calibration parameters and is computationally efficient. The model showed significant skill in reproducing the shoreline evolution during one and a half years at Nova Icaria Beach and more than sixteen years at Campo Poseidón. Both study sites showed strong seasonal variation, with slow accretion for periods of low-energy waves and faster erosion during high-energy wave events. 18
Journal Pre-proof The results were statistically significant considering that the correlation coefficient between the observed and modeled shoreline exceeded 0.6 in both study sites. In general, the small-scale erosion-accretion trend and the sediment gain or loss tendency in the medium-long term have been well represented at qualitative and quantitative levels. ACKNOWLEDGMENTS The authors would like to acknowledge the support of the Society for the Regional Development of Cantabria, SODERCAN group under Grant ID16-IN-045, SMC2020 Project, and the support of the Spanish Ministry of Economy, Industry and Competitiveness under Grant BIA2017-89491-R Beach-Art Project. The Mediterranean Center for Marine and Environmental Research (CMIMA) is kindly acknowledged for providing the pictures of Nova Icaria Beach from the “Barcelona Littoral Station”. Additionally, the authors extend their thanks to Repsol Investigaciones Petrolíferas S.A. (R.I.P.S.A.) for providing the beach profile surveys of the Campo Poseidón monitoring program.
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Journal Pre-proof
An equilibrium shoreline evolution model adding a rate of change component is presented. The relationship between the equilibrium shoreline position and the incident wave energy must evolve over time on beaches subjected to sediment deficit or surplus. The model successfully represents the shoreline evolution at two study sites on the Spanish coast, considering the progressive trend of sediment gains or losses.
Camilo Jaramillo, Martínez-Sánchez Jara, Mauricio González, Raúl Medina PII:
S0378-3839(19)30203-0
DOI:
https://doi.org/10.1016/j.coastaleng.2019.103612
Reference:
CENG 103612
To appear in:
Coastal Engineering
Received Date:
03 May 2019
Accepted Date:
22 November 2019
Please cite this article as: Camilo Jaramillo, Martínez-Sánchez Jara, Mauricio González, Raúl Medina, A SHORELINE EVOLUTION MODEL CONSIDERING THE TEMPORAL VARIABILITY OF THE BEACH PROFILE SEDIMENT VOLUME (SEDIMENT GAIN / LOSS), Coastal Engineering (2019), https://doi.org/10.1016/j.coastaleng.2019.103612
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Journal Pre-proof A SHORELINE EVOLUTION MODEL CONSIDERING THE TEMPORAL VARIABILITY OF THE BEACH PROFILE SEDIMENT VOLUME (SEDIMENT GAIN / LOSS) Camilo Jaramillo1*, Martínez-Sánchez Jara1, Mauricio González1, Raúl Medina1 1Environmental
Hydraulics Institute, Universidad de Cantabria - Avda. Isabel Torres, 15, Parque Científico y Tecnológico de Cantabria, 39011, Santander, Spain
* Corresponding author Email addresses: [email protected] (Camilo Jaramillo), [email protected] (MartínezSánchez Jara), [email protected] (Mauricio González), [email protected] (Raúl Medina)
1
Journal Pre-proof ABSTRACT The prediction of shoreline variability along sandy coastlines is a valuable skill for a broad range of coastal engineering applications. Currently, there are multiple proposals in the literature for equilibrium cross-shore shoreline evolution models based on a constant equilibrium condition correlated with the shoreline position. The aim of this paper is to present an extension of an existing equilibrium shoreline evolution model adding a rate of change component, which in turn modifies the relationship between the equilibrium shoreline position and the incident wave energy, as a function that advances or retreats over time. The proposed model is applied to two study sites on the Spanish coast. On the one hand, one and a half years of shoreline position data obtained from a video camera system at Nova Icaria Beach, Catalonia, are used to evaluate the shoreline evolution during a period of sediment contribution, which is attributed to a sand bypass coming from Bogatell Beach. On the other hand, 16 years of shoreline position data acquired from beach profile surveys in Campo Poseidón, Huelva, are used to reproduce the shoreline evolution, taking into account the trend of sediment loss due to sediment drift alongshore. The results obtained from both study sites using the proposed model show overall good performance. The model successfully represents the general shoreline erosion–accretion oscillations and the progressive trend of sediment gains or losses through the evolutionary relationship between the equilibrium shoreline position and the incident wave energy.
Key words: equilibrium shoreline evolution models, cross-shore, shoreline trend, equilibrium energy function.
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Journal Pre-proof 1. INTRODUCTION Most sandy beaches worldwide have a tendency of progressive shoreline advances or retreats for years or even decades, and this pattern is a product of multiple factors, which may be of natural or anthropic origin. In addition, the beaches usually suffer from specific alterations at a shorter time scale, which are reflected by a representative sediment deficit or surplus (Mentaschi et al., 2018). Therefore, to predict the shoreline variability in such cases, the numerical models of morphological evolution must contemplate a rate of change of shoreline position for sediment gain or loss. In the literature, there is an extensive list of different types of models that simulate shoreline evolution: multi-line shoreline models (e.g., Bakker, 1970; Perlin and Dean, 1979; 1983; Hanson and Larson, 2000), one-line shoreline models (e.g., Pelnard-Considere, 1956; Hanson and Kraus, 1991; Dabees and Kamphuis, 1998), equilibrium shoreline evolution models (e.g., Kriebel and Dean, 1993; Yates et al., 2009; Davidson et al., 2010; 2013; Jara et al., 2015) or combined models (e.g., Vitousek et al., 2017; Robinet et al., 2017; 2018). The computational cost of the processbased morphodynamic models (van Rijn et al., 2003) prevents their application to multi-annual or even inter-annual morphological changes. In contrast, equilibrium shoreline evolution models are computationally efficient and can be applied to investigate long-term morphological changes (Davidson and Turner, 2009). The equilibrium shoreline evolution models try to reproduce the shoreline evolution based on a kinetic equation in which the change rate is proportional to the difference between a theoretical equilibrium condition and the current conditions. Recently, Jara et al., (2018) presented a summary of most up-to-date literature related to this type of model, and the summary was divided into two approaches: 1. models based on an equilibrium condition correlated with the shoreline position (e.g., Yates et al., 2009; Jara et al., 2015; Doria et al., 2016) and 2. models based on an equilibrium condition as a weighted average of antecedent conditions (e.g., Davidson et al., 2013; Splinter et al., 2014). The model proposed in this paper is framed into the first group. Most likely, the most widely used model based on an equilibrium condition correlated with the shoreline position corresponds to the proposal of Yates et al., 2009 (hereafter defined as the YA09 model). The YA09 model has been successfully applied to various beaches governed by strong seasonal and interannual variability due to cross-shore sediment transport (e.g., Yates et al., 2011; Castelle et al., 2014; Lemos et al., 2018); however, these study cases did not experience a marked trend of sediment gains or losses in the medium-long term. To extend the application of the empirical equilibrium models and to account for unexplained shoreline motions, such as long-term trends that are uncorrelated with wave climate changes at small time scales (days or weeks), some authors have added a constant to the model equation. In this way, Long and Plant, (2012) proposed a shoreline change model incorporating short- and long-term evolution integrated into a data assimilation framework. They considered a linear trend term for longshore approximation; nevertheless, they followed the YA09 bases, assuming a constant relationship between the equilibrium shoreline position and the incident wave energy (this relation is known as the “equilibrium energy function”, EEF; Jara et al., 2015) for forecasting purposes. Long and Plant, (2012) tested their model in a synthetic case of 2 years of monthly sampled data, but not in any cases of real beaches. Jara et al. (2015) concluded that the relationship between the equilibrium shoreline position and the incident wave energy is implicitly related to the morphological beach characteristics; 3
Journal Pre-proof more specifically, a constant linear relationship would be associated with a constant sediment volume delimited in the active beach profile. The above relation is translated into the following consequence: if an empirical equilibrium model uses a constant EEF for a study site with a net sediment volume deficit or surplus, then the evolution model would amplify or overestimate the accretion-erosion oscillations in an unrealistic way. Therefore, a constant EEF is not appropriate for study sites subjected to significant sediment gains or losses. Based on the above, the aim of this study is to present a new extension of the YA09 model; this extension considers the variation in the EEF by means of a trend rate of sediment gain or loss, 𝑣𝑙𝑡. This rate is a source or sink of sediments related to unresolved processes, as defined by Vitousek et al., (2017). There are multiple natural or anthropic origins that may cause this trend rate on sandy beaches, such as littoral sediment drift alongshore (e.g., Komar and Inman, 1970; Miller, 1999), sediment discharge by the rivers (e.g., Baban, 1995; Boateng et al., 2012), nourishments (e.g., Hanson et al., 2002; Ludka et al., 2018; Tonnon et al., 2018), dune erosion (e.g., Castelle et al., 2017; Splinter et al., 2018), cliff retreat (e.g., Limber et al., 2018; Young, 2018), aeolian sediment transport (e.g., Davidson-Arnott and Bauer, 2009; Cohn et al., 2018) and others. This paper is organized as follows: first, the theoretical development of the proposed model is explained in Section 2. Two study sites were selected to test the model: first, we used one and a half years of shoreline position data obtained from a video camera system at Nova Icaria Beach, Catalonia; and second, we used approximately 20 years of shoreline position data acquired from beach profile surveys at Campo Poseidón, Huelva. The study sites, marine conditions and shoreline acquisition methods are detailed in Section 3. The model results for both study sites are shown in Section 4, and further discussions are provided in Section 5. Finally, the main conclusions are summarized in Section 6. 2. MODEL DEVELOPMENT As stated previously, several shoreline evolution models based on an equilibrium condition correlated with the shoreline position have been developed to simulate the shoreline position variability due to storm response and seasonal changes. These models relate the rate of cross𝑑𝑆 shore shoreline displacement, 𝑑𝑡, to the incident wave energy and the wave energy disequilibrium between the wave energy and the equilibrium wave energy that would cause no change to the present shoreline location (Castelle et al., 2014). The model proposed in the present study is based on the kinetic equation defined by Yates et al. (2009). According to the YA09 model, the governing equation that describes the rate of shoreline change is of the form (Eq. (1)): 1 𝑑𝑆(𝑡) = 𝐶 ± 𝐸 2 ∙ ∆𝐸(𝑆) 𝑑𝑡
(1) ∆𝐸 = (𝐸 ― 𝐸𝑒𝑞) (2) where S(t) and E are the shoreline position (m) and the incident wave energy (m2), respectively, at time “t”, E1/2 is the square root of the incident wave energy as an energy weighting factor that 4
Journal Pre-proof prevents nonphysical changes in the shoreline position (Yates et al., 2009), ΔE (see Eq. (2)) is the wave energy disequilibrium, C± is the proportionality constant of accretion (m-2s-1), C+, when ΔE < 0, or erosion, C-, when ΔE > 0, and Eeq is the equilibrium wave energy. The shoreline position, S, is computed as the average position of the shoreline in relation to a constant reference point or line in the backshore (see Fig. 2). Following Jara et al., (2018), the incident wave energy, E can be expressed as the zero-order moment of the wave height at breaking, Hb, according to Thornton and Guza, (1983). 𝐸=
𝐻𝑏
( )
2
4.004
(3) In both formulations (Eq. (1) and (2)), it remains to be defined how the equilibrium wave energy, Eeq, is estimated. The value of Eeq is determined in the YA09 model using a linear function of the shoreline position following the next expression: 𝐸𝑒𝑞(𝑆) = 𝑎𝑆 + 𝑏 (4) where “a” and “b” are the slope (m2/m) and the ordinate axis intercept (m2), respectively. Yates et al. (2009) defined the equilibrium shoreline position, 𝑆𝑒𝑞, for a given wave energy as follows: 𝑆𝑒𝑞 =
𝐸―𝑏 𝑎 (5)
The analytical correlation of Eq. (5) is the previously defined EEF (Jara et al., 2015). Basically, this correlation is the cornerstone of shoreline evolution models based on an equilibrium condition correlated with the shoreline position. As Jara et al. (2015) concluded, the EEF implicitly considers the sediment volume available in the physiographic unit; therefore, a constant EEF is useful for beaches that preserve their sediment volume over time. However, a constant EEF is not useful for study sites subjected to significant sediment gains or losses. Based on the aforementioned aspects, we propose an extension of the YA09 model including a linear trend term, 𝑣𝑙𝑡, for the longshore approximation (m/s) and this term should be considered in the EEF behavior while the beach under study is subjected to net sediment gains or losses. In this sense, Eq. (5) must evolve over time in such a way that the relationship is an increasing or decreasing function, depending on the rate of sediment gain/loss acting on the physiographic unit. In this study, the EEF takes the following form:
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𝐸―𝑏 + 𝑣𝑙𝑡 ∙ 𝑡 𝑎 (6)
It is important to note that Long and Plant, (2012) (hereafter referred to as the LP12 model), included the linear trend term, 𝑣𝑙𝑡, to the YA09 model, but in the Eq. (1) adding also the Kalman filter. This data assimilation method automatically adjusts the model parameters (a, b, C± and 𝑣𝑙𝑡 ) during runtime to best fit any available observed shoreline data at the concurrent time step (Vitousek et al. 2017). In other words, as concluded by Vitousek et al. (2017), the Kalman filter is used as a largescale calibration tool, and after the last available shoreline observation is assimilated, the model parameters remain constant for the entire forecasting period. From the above, it is inferred that the LP12 model, including the Kalman filter, uses an EEF that evolves during the calibration period, but then, it remains constant for the forecasting period. Fig. 1 shows an EEF example of the proposed model based on the study site of Nova Icaria Beach, which is evaluated in detail in Section 4.1. The upper panel of Fig. 1 shows only one month of time series of incident wave energy, and the bottom panel shows the corresponding empirical approximation of the EEF (black line). The points are consecutive measurements that are colored according to the shoreline change rate.
Fig. 1. One month of incident wave energy at Nova Icaria Beach (upper panel) as driver of an EEF (black line in bottom panel). The points are consecutive measurements that are colored according to the shoreline change rate.
2.1. Model hypotheses and assumptions The proposed model in this study assumes the same bases as the YA09 model, in terms that shoreline changes mainly respond to wave energy and it is insensitive to wave direction, and as such, it is best suited for locations where waves are the primary driver of shoreline response.
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Journal Pre-proof Also, the model does not account for short-scale processes such as beach cusp formation, alongshore variable bar welding, or rip current embayments (Splinter et al., 2014). By neglecting the wave directionality of each sea state, the model is not able to capture the short-term variability of the longshore sediment transport. However, the long-term shoreline change due to gradients in longshore transport and/or onshore/offshore feeding/loss of sand may be captured by the linear trend term, 𝑣𝑙𝑡. This linear term has been included to consider the overall longshore processes, considering that where these processes cannot be captured by the linear term (or the wave-driven model component), the model does not accurately resolve the shoreline response. Following the conclusions found by Yates et al., (2009); (2011) the sediment grain size affect the magnitude of the equilibrium slope “a” and the rate change coefficients C±, both of which affect the rate of beach change. From the above, variations of free parameter values between sites are implicitly associated with the beach physical characteristics. In this study, it is considered that the calibration parameters (C- and C+) associated to erosion and accumulation processes remain constant during the entire model simulation. With respect to the tidal range, the model does not explicitly include any additional parameter. Nonetheless, as suggested by Castelle et al., (2014) the equilibrium shoreline evolution models can be applied to a range of elevation contours in the intertidal zone to evaluate the effect of wave energy at different altitudes along the intertidal beach profile, with satisfactory efficiency. Castelle et al., (2014) concluded that the best shoreline proxy in their mesotidal study site was related to the mean high water level, where the inner-bar and berm dynamics have little influence on the shoreline cross-shore displacement. This conclusion was supported by Lemos et al., (2018) who analyzed a macrotidal environment and found that the equilibrium models shows a good predictive ability in the upper intertidal zone where the sediment dynamics depend mainly on the waves energy. As suggested by Splinter et al., (2014), the exclusion of water level precludes the impacts of changes in mean water level due to climatological impacts, such as storm surge, El Niño Southern Oscillation (ENSO), and sea level rise. 2.2. Model implementation From the governing equations described above, this section presents a brief methodology to apply the proposed model. The first step is to identify whether the beach under study is subjected to a net gain or loss of sediments. For this, the slope of the trend term, 𝑣𝑙𝑡, relies on a regression of historical data as long as possible. This term can be obtained by historical records of shoreline surveys or, for example, by evaluating the average rate of alongshore sediment transport by means of historical wave data propagated to the coast. The next step is to determine the remaining calibration parameters (a, b and C±). For this, it is necessary to define a calibration time period in which a constant EEF is evaluated by means of Eqs. (1), (2) and (6), considering that 𝑣𝑙𝑡 = 0. This initial relationship is the best fit line to the observed average wave energy causing no change in the shoreline position. The calibration parameters C± must be obtained through an iterative algorithm (see Section 4). The calibration period can be considered as follows:
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A period of time in which there is no significant trend of a gain or loss of sediments. A period of time with a significant trend of a gain or loss of sediments. In this case, the trend term, 𝑣𝑙𝑡, must be filtered.
Once all the calibration parameters (a, b, C± and 𝑣𝑙𝑡) have been defined, the model is validated using Eqs. (1), (2) and (6), considering that 𝑣𝑙𝑡 ≠ 0. In summary, the EEF starts from an initial linear relationship from which it evolves according to the rate of sediment gain/loss acting on the beach. 3. STUDY SITES In this section, the selected study sites are presented. The right panel of Fig. 2 corresponds to Nova Icaria Beach, as this represents a study site where the beach accretion process was experienced in a limited time period (Ojeda and Guillén, 2006); on the left panel of Fig. 2, there is a monitored beach profile of Rompeculos Beach (referred to as the Campo Poseidón beach profile), where a continuous sediment loss process has been recorded (Fernández et al., 1990).
Fig. 2. Location of the selected study sites. Campo Poseidón (lower and left panel), including the “Downscaled Ocean Waves” (DOW) position, Tide Gauge Huelva and monitoring beach profile. Nova Icaria Beach (upper and right panel), including the DOW position, Tide Gauge Barcelona and the ARGUS station with the corresponding Camera C5. In both cases the reference point or line to estimate the shoreline position, S, is indicated.
The marine conditions acting in the vicinity of both study sites (see Sections 3.1.2 and 3.2.2) are characterized by the incoming waves acquired from the “Downscaled Ocean Waves” (DOW) (Camus et al., 2013) and the sea levels recorded by the closest tide gauges. The DOW database is a historical reconstruction of coastal waves by means of a hybrid methodology based on both dynamical and statistical downscaling for a period of 68 years (1948 to 2015) along the Spanish coast; this database provides hourly sea states with different wave parameters (e.g., significant wave height, Hs, peak period, Tp, mean wave direction, θm). Specifically, the DOW is a downscaled wave reanalysis of coastal zones from a Global Ocean 8
Journal Pre-proof Waves (GOW) database (Reguero et al., 2012), which considers a correction of open sea significant wave height through a directional calibration (Mínguez et al., 2011) and has been validated with records from the closest buoys. Regarding sea levels, the Spanish Grid of Gauges (REDMAR) provides data for astronomical tide levels and surge tide levels. The selected tide gauges correspond to the Port of Barcelona and the Port of Huelva for Nova Icaria Beach and Campo Poseidón, respectively.
3.1. Nova Icaria Beach 3.1.1. Study site Nova Icaria Beach is a 400 m-long non-barred sandy beach located at Barcelona (Catalonia), northeastern Spain (see Fig. 2). The beach is embedded between the Olympic Port of Barcelona to the south and the Bogatell dike to the north. The beach is also protected by a semi-submerged curved breakwater, which is an extension of the Bogatell dike (Z = 0 m with respect to the MSL), and a small breakwater that protects the boat-access ramp in the northeastern section of the Olympic Port. The average direction normal to the beach orientation is close to 135° with respect to north, and the beach sediment is characterized by an average median grain size (D50) of approximately 0.43 mm (Ojeda and Guillén, 2008). Nova Icaria Beach is one of the most intensively studied coastal sites in Catalonia (e.g., Ojeda and Guillén, 2006; Guillén et al., 2008; Ribas et al., 2010; Ojeda et al., 2011; Turki et al., 2012; 2013; Jara et al., 2015), and its historical morphodynamic variability has been reported in detail. In this study, it is worth highlighting the shoreline monitoring analysis of Barcelona's beaches developed by Ojeda and Guillén (2006) and Guillén et al. (2008), who identified a process of sediment bypass from Bogatell Beach to Nova Icaria Beach. This event occurred months after nourishment works at Bogatell Beach were carried out between 13th June and 5th July 2002, with a sediment contribution of approximately 70,000 m3. It is important to highlight that headland bypassing due to moving sand around Bogatell dike (see Fig. 2) was initially reported by Peña and Covarsl (1994), who used drift monitoring analysis using transects. This sediment drift was controlled in 2010, when the construction of a submerged barrier was implemented to delimit the sand contained in the physiographic unit of Bogatell Beach. 3.1.2. Marine conditions The Catalan coast (Western Mediterranean) is a microtidal region where the astronomical tide is characterized by a semi-diurnal regime (mean range of 0.23 m), and storm surges can reach values up to 0.5 m greater than the astronomically predicted tide level. In this region, the effect of tides on the morphology of the beaches is significantly less than waves. The selected DOW point near Nova Icaria Beach is located at latitude 41.387° and longitude 2.207°, with a water depth of 13.3 m (see Fig. 2). This point has been validated with measurements of an OPPE (Organismo de Puertos del Estado) buoy at Barcelona. Fig. 3a shows the scatter diagram that correlates the significant wave height, Hs, with the peak period, Tp, and Fig. 3b depicts the directional rose of the Hs.
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Fig. 3. Wave characteristics of DOW point at Nova Icaria Beach. a) Scatter diagram Hs-Tp. b) Directional rose of Hs.
A statistical analysis of the wave conditions from 1948 to 2015 shows that the dominant or more frequent waves and the most energetic waves come from the ESE (24% of the time), followed by waves coming from the E (21.9%) and SE (21.7%). The Hs and Tp have ranges of 0.01–5 m and 1–13.4 s, respectively. The wave height exceeded 50% of the time is 0.41 m, and the significant wave height exceeded 12 hours per year (Hs12) is approximately 3 m. The significant wave height time series presents a cyclical behavior, with storm periods (October - April) separated by periods of low storm activity (May - October). 3.1.3. Shoreline data Daily shoreline records during the time period, which spanned from May 2002 to October 2003, were obtained from snap images taken by the Argus video system “Barcelona Littoral Station” located on top of the Mapfre Building by the Olympic Marina approximately 142 m above mean sea level (MSL). This Argus system consists of five video cameras operated by the Mediterranean Center for Marine and Environmental Research (CMIMA), and these cameras have a 180° view of the coast. The shorelines were manually plotted and digitized over oblique pictures taken by camera C5 and then rectified by means of a direct linear transformation technique, with a maximal expected error of 1 m at the northern tip of the beach (Jara et al. 2015). Fig. 4a shows an example of a rectified snap picture including two shorelines associated with the summers before and after the bypass period (approximately from October 2002 to June 2003) from Bogatell Beach to Nova Icaria Beach. Fig. 4b shows the time-series evolution of the mean cross-shore position.
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Fig. 4. Shoreline position at Nova Icaria Beach. a) Example of rectified snap picture showing two surveys: the blue line is the shoreline associated with the day 1507//2002 (before bypass), and the red line is the shoreline associated with 15/07/2003 (after bypass). b) Time-series of shoreline positions.
The shoreline position, S, for the entire beach has been computed as the mean position in relation to the constant reference line in the backshore (see Fig. 4a). The shoreline reference level corresponds to the local MSL. To minimize the possible overestimation of shoreline changes due to storm surges and wave set-up, the shoreline positions extracted from rectified pictures have been corrected with the MSL position at the moment the picture was captured, considering the astronomical tide, storm surge and wave set-up. 3.2. Campo Poseidón 3.2.1. Study site The second study site corresponded to the Campo Poseidón beach profile (see Fig. 2). This monitoring section is located at Rompeculos Beach in Moguer municipality (Huelva), on the southwest coast of Spain. It is a coastline stretching 3000 m-long and encompasses a long sandy beach (more than 50 km) bounded by the Port of Mazagón and the delta of the Guadalquivir River. The average direction normal to the Rompeculos Beach orientation is close to 210° with respect to the north, and the beach sediment is characterized by D50=0.3 mm (Fernández et al., 1992). The east coast of Huelva has a long history of monitoring programs, which have shown how highly recessive this area is. Fernández et al. (1990) quantified the coastline retreat at a rate of 1.5 m/yr over 30 years (since the 1960s). This value is consistent with the estimated trend from data measured over more than 16 years (December 1998 to July 2015) at the Campo Poseidón beach profile (see Section 4.2). As Medina et al. (1992) concluded, the coastal retreat of this region has been caused by two different reasons: first, the littoral drift from west to east, and
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Journal Pre-proof second, the reduction in the volume of sand transported by the rivers to the coast mainly caused by human actions (e.g., construction of dams, littoral barriers). 3.2.2. Marine conditions The Gulf of Cadiz faces the Atlantic Ocean and is characterized by a mesotidal regime with a medium neap to spring variation (1.20–3.30 m) (Muñoz-Perez et al., 2001). According to datum references of the Tide Gauge of Port of Huelva, the mean lower low water (MLLW) level is 0.47 m, the MSL is 2.03 m and the mean higher high water level (MHHL) is 3.63 m, all of which are above the reference level or zero of the port. The storm surges can reach values up to 0.6 m greater than the astronomically predicted tide level. The selected DOW point near Campo Poseidón is located at latitude 37.1° and longitude 6.765°, with a water depth of 6.3 m (see Fig. 2). This point has been validated with measurements of OPPE-Sevilla. Fig. 5a shows the scatter diagram Hs-Tp, and Fig. 5b shows the directional rose of the Hs.
Fig. 5. Wave characteristics of DOW point at Campo Poseidón. a) Scatter diagram of Hs-Tp. b) Directional rose of Hs.
The statistical analysis of wave conditions from 1948 to 2015 shows that the most frequent and most energetic waves come from the SW (24% of the time), followed by waves from the S (19%) and WSW (18%). The Hs and Tp have ranges of 0.01–4 m and 1–19 s, respectively. The wave climate is strongly seasonally modulated, with an annual mean significant wave height, Hs, of 0.39 m and an Hs12 of approximately 2.7 m. The winter is the most energetic period (i.e., December-January). 3.2.3. Shoreline data Repsol Investigaciones Petrolíferas S.A. (R.I.P.S.A.) has monitored the Campo Poseidón beach profile since December 1998. From the beginning until 2013, 10 campaigns per year of intertidal beach profile surveys were carried out, and these sampling were monthly except in the months of July and August. From 2013 onwards, the program was extended to 12 measurements per year, including profiles for July and August. Following the research developed by Castelle et al., (2014) and Lemos et al., (2018), the intersection of the coastal profile with the vertical elevation MHHL was selected as a shoreline proxy. According to these authors, the shoreline cross-shore displacement related to the MHHL contour is slightly influenced by the inner-bar and berm dynamics. Based on the above information, Fig. 6 shows the time-series evolution of the mean cross-shore position associated with the MHHL contour.
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Fig. 6. Time-series of shoreline positions at the Campo Poseidón beach profile considering the MHHL contour as the reference level.
4. RESULTS This section presents the performance of the proposed equilibrium cross-shore shoreline evolution model applied to the two selected study sites. In both cases, the time series of waves at the DOW points were propagated to the breaking point using linear wave theory and the Thornton and Guza (1983) linear breaking criterion. The wave breaking conditions have been used to obtain the incident wave energy, which is the main model driver. The calibration procedure consists of finding the best model fit using an error minimizing technique to solve the calibration parameters. To do this, an iteration algorithm was used to find the proportionality constants of accretion, C+, and erosion, C-, that produced the lowest root mean square error (RMSE; Eq. (7)) between the modeled and measured shoreline positions. 𝑛
𝑅𝑀𝑆𝐸 =
∑𝑖 = 1(𝑆𝑑𝑎𝑡𝑎,𝑖 ― 𝑆𝑚𝑜𝑑𝑒𝑙, 𝑖)2 𝑛
(7) where i and N represent each datum and the total size of the sample, respectively. Section 4.1 presents the results from Nova Icaria Beach, and Section 4.2 presents the results from Campo Poseidón. 4.1. Model results from Nova Icaria Jara et al. (2015) analyzed the shoreline evolution of Nova Icaria Beach during a natural beach oscillation time period that spanned from January 2005 to January 2007 using two equilibrium cross-shore shoreline evolution models. They successfully calibrated and validated their proposed model (named the JA15 model) and the YA09 model. From that research, the present study inherits the YA09 model calibration parameters obtained from the six-month period between January and June 2005 (a=-0.068106 m2/m, b=6.13 m2, C+=-2.4·10-4 m-2s-1, C-=-1.1·104 m-2s-1). It should be noted that the optimal erosion and accretion change rate coefficients, C±, are of the same order of magnitude. The proposed model has been validated for a time period of one and a half years (see Fig. 7d). The first and last five months correspond to the natural oscillations of Nova Icaria Beach before and after the sediment bypass period (approximately from October 2002 to June 2003), respectively (Ojeda and Guillén, 2006).
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Journal Pre-proof During the sediment bypass period (gray shaded area in Fig. 7d), the shoreline position of Nova Icaria Beach advanced at a rate of 11.55 m/yr (see Fig. 7a). Fig. 7b shows the EEF that defines the equilibrium shoreline position based on the incident wave energy. It is important to highlight that the beginning of the EEF (to the left in this case) is applicable to Nova Icaria Beach’s situation before the sediment bypass, while the end of the function governs the beach variability after the sediment bypass.
Fig. 7. Results for Nova Icaria Beach. a) Linear fit to the shoreline evolution during the bypass period. b) EEF (black line) considering the complete study period. c) Time series of incident wave energy at the breaking point. d) Validation of the proposed shoreline evolution model.
According to the results presented in Fig. 7d, the proposed cross-shore shoreline equilibrium model including the trend term allows the simulation of the progressive shoreline advance in accordance with the sediment supply contributed during the bypass period from Bogatell Beach to Nova Icaria Beach. In addition, the model exhibits robust behavior when simulating the general shoreline advances and retreats. Both observed and modeled shoreline positions show fast erosion after major storms and slower accretion in the post-storm recovery period. For example, given the time series of incident wave energy at the breaking point (see Fig. 7c), a storm of greater intensity occurred in mid-October 2003; this storm is reflected by a strong shoreline retreat (Fig. 7d). The resulting RMSE between the observed and modeled shoreline positions is 1.62 m, which is similar to the expected error due to the image rectification (1 m at the northern tip of the pocket beach) and to the results obtained by Jara et al. (2015) for the study period of 2005-2007. The correlation coefficient, ρ, between the observed and modeled shoreline positions is 0.87, which is a clear result indicating a strong correlation. 4.2. Model results from Campo Poseidón 14
Journal Pre-proof In the case of the Campo Poseidón beach profile, the first 3 years (December 1998-December 2001) of shoreline measurements were considered for calibration purposes (see gray shaded area in Fig. 8d). Because the beach experiences a continuous retreat at a rate of 1.5 m/yr according to Fernández et al. (1990), the calibration period must filter this trend to obtain the model calibration parameters (a, b and C±). The iteration technique mentioned at the beginning of Section 4 was used to find the best set of free parameters that minimize the RMSE between the modeled and observed shorelines. The resulting calibration parameters are a=-0.0027 m2/m, b=0.5595 m2, C+=-5.7·10-4 m-2s-1 and C-=-9.7·10-5 m-2s-1. Then, the proposed model was validated for more than thirteen years of shoreline measurements (from January 2002 until July 2015), as shown in Fig. 8d. Although the shoreline position record obtained from the beach profile surveys is approximately 20 years (see Section 3.2.3), the simulation was run until mid-2015 based on the wave data availability (see Section 3.2.2). During the study period, the shoreline position at the Campo Poseidón beach profile retreated at a rate of -1.55 m/yr (see Fig. 8a). In this case, the beginning of the EEF (to the right in Fig. 8b) is related to the linear relationship (E-S) defined for the calibration procedure, and then, the EEF evolves based on the sediment loss rate.
Fig. 8. Results at Campo Poseidón. a) Linear fit to the shoreline evolution. b) EEF (black line) considering the complete study period. c) Time series of incident wave energy at the breaking point. d) Calibration (gray shaded period) and validation of the proposed shoreline evolution model.
Fig. 8c shows a strong seasonal variation of the incident wave energy, with strong storms during winter periods and low energy waves during the summers. The results presented in Fig. 8b reveal that the model, again, has achieved good performance, successfully reproducing the general advance/retreat shoreline trend. The resulting RMSE is 10.4 m, and the correlation coefficient,
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Journal Pre-proof ρ, between the observed and modeled shoreline positions is 0.64, which is a satisfactory correlation. 5. DISCUSSION According to the results presented in Section 4, the proposed model showed significant skill in reproducing the shoreline evolution at two study sites with different characteristics: 1. Nova Icaria Beach was subjected to sediment supply in a microtidal environment, and 2. Campo Poseidón was subjected to sediment loss in a mesotidal environment. In addition, the data sources used to obtain the time series of the shoreline positions were different in both cases; in the first, data came from video-camera images, and in the second, data came from beach profile surveys. The study site of Nova Icaria Beach has been used for direct model validation because the sediment bypass period from Bogatell Beach was defined a priori, and the calibration parameters were inherited from Jara et al. (2015). During the first five months, the model uses a linear EEF with 𝑣𝑙𝑡=0 m/yr; then, the EEF evolves during the bypass period at the rate of sediment increase, and it finally continues with the last linear EFF and 𝑣𝑙𝑡=0 m/yr for the next five months. On the other hand, the study site of Campo Poseidón has been completely predictive, and the model has achieved an overall good performance, simulating nearly 16 years of shoreline evolution. In this case, the value of the loss rate recorded with historical data (𝑣𝑙𝑡 =1.5 m/yr) by Fernández et al. (1990) has been considered, and it has been applied to the longterm in a continuous erosion trend, which results in a recessive EEF. To assess the advantages of the proposed model, Fig. 9 shows a comparative model simulation for the two study sites considering two scenarios: 1. Model includes trend term (same results of Section 4), and 2. Model establishes 𝑣𝑙𝑡=0 m/yr during the entire study period. Table 1 summarizes the values of the mean squared error and correlation coefficient found in this comparative analysis.
Fig. 9. Comparison of results using the proposed model including the trend term versus the model without the trend term at a) Nova Icaria Beach and b) Campo Poseidón.
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Journal Pre-proof Table 1. Root mean square error and correlation coefficient between the observed and modeled shoreline positions at Nova Icaria Beach and Campo Poseidón considering two scenarios.
Nova Icaria Beach 𝒗𝒍𝒕 (m/yr) RMSE (m) ρ Without trend term 5.18 0.08 n/a With trend term 1.64 0.87 11.55 Model
Campo Poseidón 𝒗𝒍𝒕 (m/yr) RMSE (m) ρ 18.7 0.31 n/a 10.4 0.64 -1.5
According to Fig. 9, as expected, when the equilibrium shoreline evolution model does not consider the trend term, it tries to reproduce the shoreline advancing and retreating oscillations around an average position over time; however, the model does not consider net volume changes on the beach. The values presented in Table 1 reflect the weak correlation between the observed and modeled shoreline positions when the model does not include the trend term. Another test that should be highlighted is what would happened if the linear trend term were added after the YA09 model results (similar to LP12 model), but considering that the EEF is constant over time (not including the rate parameter). This assumption is represented in Fig. 10, for the case of Nova Icaria Beach. As it is observed, it would be feasible to find a good result by adding the linear change rate during the bypass period; however, if the model assumes a constant EEF, the simulation will tend to the established equilibrium condition, from which, the model is not able to assimilate that the beach has changed its sediment volume over time.
Fig. 10. Comparison between the proposed model and a simulation considering the linear trend term added after the YA09 model result (bypass period), but assuming that the EEF is constant over time (bottom panel). Differences in meters between both simulations (upper panel).
During the bypass period, only differences less than 25 cm are registered (see upper panel of Fig. 10). These slight variations of erosion/accretion fluctuations are explained in the differences between the kinetic equations of both cases, presented in Table 2. Table 2. Comparison of kinetic formulations between the proposed model versus YA09 model results adding linearly the long term.
YA09 model results adding linearly the long term
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Proposed model
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[
1 𝑑𝑆(𝑡) = 𝐶 ± 𝐸 2(𝐸 ― 𝐸𝑒𝑞(𝑆)) + 𝑣𝑙𝑡 ∙ 𝑡 𝑑𝑡
]
1 𝑑𝑆(𝑡) = 𝐶 ± 𝐸 2(𝐸 ― 𝐸𝑒𝑞(𝑆)) 𝑑𝑡
(10)
(8) 𝐸𝑒𝑞(𝑆) = 𝑎(𝑆 ― 𝑣𝑙𝑡 ∙ 𝑡) + 𝑏
𝐸𝑒𝑞(𝑆) = 𝑎𝑆 + 𝑏
(9)
(11)
It should be noted that the proposed model assumes that 𝑣𝑙𝑡 is linearly constant during the forecasting period; nonetheless, as indicated in the Introduction, in reality, there are multiple sources or sinks of sediments on beaches (e.g., littoral sediment drift alongshore, nourishments, cliff retreat) and they are variable over time. Therefore, assuming a constant rate, the proposed model captures only the general tendency, but the nonlinear changes cannot be precisely resolved. For example, the month of February 2003 in the simulated results of Nova Icaria Beach and the years 2010 and 2012 in the simulated results of Campo Poseidón had more advanced shoreline position records than did the model predictions. Note that in those periods where the model is not able to replicate the shoreline response, it should be also due to the others assumptions described in detail in Section 2.1 (not consideration of wave directionality, changes in the wave-driven model component speed or changes of the physical beach parameters). In order to make the term 𝑣𝑙𝑡 and the other parameters (a, b, C+ and C-) variable over the time, it would be possible to include a data assimilation algorithm, such as the Kalman Filter, previously proposed by Long and Plant, (2012). This algorithm would automatically adjusts the rate parameters during the runtime to best fit any available observed data at the concurrent time step. However, after the last available observation is assimilated, the parameters will remain constant for the entire forecasting period. The limitations inherited by assuming a linear trend constant suggest that the proposed model could even improve its prediction by including a better prediction of the source or sink variability, e.g., a littoral sediment drift alongshore function obtained from historical seasonal analysis or a sediment discharge function associated with streamflow hydrographs of rivers, among others. Finally, time series of shoreline measurements in different sites with diverse beach characteristics and wave conditions would be useful to widely validate the robustness of the shoreline evolution model, including the analytical form of the EEF proposed in this paper. 6. CONCLUSIONS This present study proposes an extension of the equilibrium shoreline evolution model originally developed by Yates et al., (2009). The new model adds a trend parameter due to unresolved processes (constant rate of sediment gain or loss), which in turn modifies the EEF as a pathway that advances or retreats over time. The model is capable of reproducing the shoreline response due to cross-shore forcing over a variety of temporal scales. It is a reduced-complexity empirical evolution model that requires few calibration parameters and is computationally efficient. The model showed significant skill in reproducing the shoreline evolution during one and a half years at Nova Icaria Beach and more than sixteen years at Campo Poseidón. Both study sites showed strong seasonal variation, with slow accretion for periods of low-energy waves and faster erosion during high-energy wave events. 18
Journal Pre-proof The results were statistically significant considering that the correlation coefficient between the observed and modeled shoreline exceeded 0.6 in both study sites. In general, the small-scale erosion-accretion trend and the sediment gain or loss tendency in the medium-long term have been well represented at qualitative and quantitative levels. ACKNOWLEDGMENTS The authors would like to acknowledge the support of the Society for the Regional Development of Cantabria, SODERCAN group under Grant ID16-IN-045, SMC2020 Project, and the support of the Spanish Ministry of Economy, Industry and Competitiveness under Grant BIA2017-89491-R Beach-Art Project. The Mediterranean Center for Marine and Environmental Research (CMIMA) is kindly acknowledged for providing the pictures of Nova Icaria Beach from the “Barcelona Littoral Station”. Additionally, the authors extend their thanks to Repsol Investigaciones Petrolíferas S.A. (R.I.P.S.A.) for providing the beach profile surveys of the Campo Poseidón monitoring program.
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An equilibrium shoreline evolution model adding a rate of change component is presented. The relationship between the equilibrium shoreline position and the incident wave energy must evolve over time on beaches subjected to sediment deficit or surplus. The model successfully represents the shoreline evolution at two study sites on the Spanish coast, considering the progressive trend of sediment gains or losses.